# Approximating functions with Taylor polynomials and error bounds

##### Intros

###### Lessons

##### Examples

###### Lessons

- Approximate ln 2 using the 3'rd degree Taylor Polynomial. Find the error term.
- Find the 4th degree Taylor Polynomial centred around $a=0$ of $f(x)=e^x$. Then approximate $e^2$.
- Find the 2nd degree Taylor Polynomial centred around $a=1$ of $f(x)=\sqrt{(x+1)}$ and the error term where $x \in [0,2]$.
- Show that $f(x)=e^x$ can be represented as a Taylor series at $a=0$.
- Show that $f(x)= \cos ?x$ can be represented as a Taylor series at $a=0$.

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###### Topic Notes

To approximate a function with a Taylor Polynomial of n'th degree centred around $a=0$, use

$f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$

where $P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$ is the Taylor Polynomial.

To find the difference between the actual value and your approximated value, look for the error term, which is defined as

$R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}$

Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,

$f(x)=P_n(x)+R_n(x)$

$f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$

where $P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$ is the Taylor Polynomial.

To find the difference between the actual value and your approximated value, look for the error term, which is defined as

$R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}$

Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,

$f(x)=P_n(x)+R_n(x)$

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